\(\int \frac {(a+b \log (c x^n))^2}{x^2 (d+e x)^4} \, dx\) [119]

Optimal result

Integrand size = 23, antiderivative size = 420 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx=-\frac {2 b^2 n^2}{d^4 x}-\frac {b^2 e n^2}{3 d^4 (d+e x)}-\frac {b^2 e n^2 \log (x)}{3 d^5}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac {8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac {3 b^2 e n^2 \log (d+e x)}{d^5}-\frac {26 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^5}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^5}-\frac {26 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 d^5}+\frac {8 b e n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^5}-\frac {8 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^5} \] Output:

-2*b^2*n^2/d^4/x-1/3*b^2*e*n^2/d^4/(e*x+d)-1/3*b^2*e*n^2*ln(x)/d^5-2*b*n*( 
a+b*ln(c*x^n))/d^4/x+1/3*b*e*n*(a+b*ln(c*x^n))/d^3/(e*x+d)^2-8/3*b*e^2*n*x 
*(a+b*ln(c*x^n))/d^5/(e*x+d)+4/3*e*(a+b*ln(c*x^n))^2/d^5-(a+b*ln(c*x^n))^2 
/d^4/x-1/3*e*(a+b*ln(c*x^n))^2/d^2/(e*x+d)^3-e*(a+b*ln(c*x^n))^2/d^3/(e*x+ 
d)^2+3*e^2*x*(a+b*ln(c*x^n))^2/d^5/(e*x+d)-4/3*e*(a+b*ln(c*x^n))^3/b/d^5/n 
+3*b^2*e*n^2*ln(e*x+d)/d^5-26/3*b*e*n*(a+b*ln(c*x^n))*ln(1+e*x/d)/d^5+4*e* 
(a+b*ln(c*x^n))^2*ln(1+e*x/d)/d^5-26/3*b^2*e*n^2*polylog(2,-e*x/d)/d^5+8*b 
*e*n*(a+b*ln(c*x^n))*polylog(2,-e*x/d)/d^5-8*b^2*e*n^2*polylog(3,-e*x/d)/d 
^5
 

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx=-\frac {\frac {6 b^2 d n^2}{x}+\frac {6 b d n \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b d^2 e n \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac {8 b d e n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-13 e \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {d^3 e \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}+\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {9 d e \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{b n}+8 b^2 e n^2 (\log (x)-\log (d+e x))+\frac {b^2 e n^2 (d+(d+e x) \log (x)-(d+e x) \log (d+e x))}{d+e x}+26 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-12 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+26 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-24 b e n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+24 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{3 d^5} \] Input:

Integrate[(a + b*Log[c*x^n])^2/(x^2*(d + e*x)^4),x]
 

Output:

-1/3*((6*b^2*d*n^2)/x + (6*b*d*n*(a + b*Log[c*x^n]))/x - (b*d^2*e*n*(a + b 
*Log[c*x^n]))/(d + e*x)^2 - (8*b*d*e*n*(a + b*Log[c*x^n]))/(d + e*x) - 13* 
e*(a + b*Log[c*x^n])^2 + (3*d*(a + b*Log[c*x^n])^2)/x + (d^3*e*(a + b*Log[ 
c*x^n])^2)/(d + e*x)^3 + (3*d^2*e*(a + b*Log[c*x^n])^2)/(d + e*x)^2 + (9*d 
*e*(a + b*Log[c*x^n])^2)/(d + e*x) + (4*e*(a + b*Log[c*x^n])^3)/(b*n) + 8* 
b^2*e*n^2*(Log[x] - Log[d + e*x]) + (b^2*e*n^2*(d + (d + e*x)*Log[x] - (d 
+ e*x)*Log[d + e*x]))/(d + e*x) + 26*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x 
)/d] - 12*e*(a + b*Log[c*x^n])^2*Log[1 + (e*x)/d] + 26*b^2*e*n^2*PolyLog[2 
, -((e*x)/d)] - 24*b*e*n*(a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] + 24*b^ 
2*e*n^2*PolyLog[3, -((e*x)/d)])/d^5
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2795, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx\)

\(\Big \downarrow \) 2795

\(\displaystyle \int \left (\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)^2}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2}{d^4 x (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x^2}+\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^3}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^2 (d+e x)^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac {8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}-\frac {8 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}+\frac {4 e \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^5}-\frac {8 b e n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^5}-\frac {6 b e n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}+\frac {8 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{3 d^5}-\frac {6 b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^5}-\frac {8 b^2 e n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^5}-\frac {b^2 e n^2 \log (x)}{3 d^5}+\frac {3 b^2 e n^2 \log (d+e x)}{d^5}-\frac {b^2 e n^2}{3 d^4 (d+e x)}-\frac {2 b^2 n^2}{d^4 x}\)

Input:

Int[(a + b*Log[c*x^n])^2/(x^2*(d + e*x)^4),x]
 

Output:

(-2*b^2*n^2)/(d^4*x) - (b^2*e*n^2)/(3*d^4*(d + e*x)) - (b^2*e*n^2*Log[x])/ 
(3*d^5) - (2*b*n*(a + b*Log[c*x^n]))/(d^4*x) + (b*e*n*(a + b*Log[c*x^n]))/ 
(3*d^3*(d + e*x)^2) - (8*b*e^2*n*x*(a + b*Log[c*x^n]))/(3*d^5*(d + e*x)) - 
 (8*b*e*n*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/(3*d^5) - (a + b*Log[c*x^n] 
)^2/(d^4*x) - (e*(a + b*Log[c*x^n])^2)/(3*d^2*(d + e*x)^3) - (e*(a + b*Log 
[c*x^n])^2)/(d^3*(d + e*x)^2) + (3*e^2*x*(a + b*Log[c*x^n])^2)/(d^5*(d + e 
*x)) + (4*e*Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d^5 + (3*b^2*e*n^2*Log[ 
d + e*x])/d^5 - (6*b*e*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/d^5 + (8*b^2 
*e*n^2*PolyLog[2, -(d/(e*x))])/(3*d^5) - (8*b*e*n*(a + b*Log[c*x^n])*PolyL 
og[2, -(d/(e*x))])/d^5 - (6*b^2*e*n^2*PolyLog[2, -((e*x)/d)])/d^5 - (8*b^2 
*e*n^2*PolyLog[3, -(d/(e*x))])/d^5
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2795

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ 
c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b 
, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 
] && IntegerQ[m] && IntegerQ[r]))
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.78 (sec) , antiderivative size = 1015, normalized size of antiderivative = 2.42

method	result	size
risch	\(\text {Expression too large to display}\)	\(1015\)

Input:

int((a+b*ln(c*x^n))^2/x^2/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

4*b^2*ln(x^n)^2/d^5*e*ln(e*x+d)-2*b^2*n*ln(x^n)/d^4/x+1/4*(I*Pi*b*csgn(I*x 
^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn 
(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a)^2*(-1/3/d^2/(e 
*x+d)^3*e+4/d^5*e*ln(e*x+d)-3/d^4*e/(e*x+d)-1/d^3/(e*x+d)^2*e-1/d^4/x-4/d^ 
5*e*ln(x))-b^2*ln(x^n)^2/d^4/x+8*b^2/d^5*e*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^2- 
8*b^2*n/d^5*e*ln(x^n)*ln(e*x+d)*ln(-e*x/d)-26/3*b^2*n*ln(x^n)/d^5*e*ln(e*x 
+d)+4*b^2*n/d^5*e*ln(x^n)*ln(x)^2+8*b^2/d^5*e*ln(x)*dilog(-e*x/d)*n^2-8*b^ 
2*n/d^5*e*ln(x^n)*dilog(-e*x/d)-4*b^2/d^5*e*n^2*ln(e*x+d)*ln(x)^2+4*b^2/d^ 
5*e*n^2*ln(x)^2*ln(1+e*x/d)+8*b^2/d^5*e*n^2*ln(x)*polylog(2,-e*x/d)+26/3*b 
^2*n*ln(x^n)/d^5*e*ln(x)+26/3*b^2/d^5*n^2*e*ln(e*x+d)*ln(-e*x/d)-4*b^2*ln( 
x^n)^2/d^5*e*ln(x)-4/3*b^2/d^5*e*ln(x)^3*n^2-13/3*b^2/d^5*n^2*e*ln(x)^2+26 
/3*b^2/d^5*n^2*e*dilog(-e*x/d)+8/3*b^2*n*ln(x^n)/d^4*e/(e*x+d)+1/3*b^2*n*l 
n(x^n)/d^3/(e*x+d)^2*e-1/3*b^2*e*n^2/d^4/(e*x+d)+(I*Pi*b*csgn(I*x^n)*csgn( 
I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n) 
^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a)*b*(-1/3*ln(x^n)/d^2/(e* 
x+d)^3*e+4*ln(x^n)/d^5*e*ln(e*x+d)-3*ln(x^n)/d^4*e/(e*x+d)-ln(x^n)/d^3/(e* 
x+d)^2*e-ln(x^n)/d^4/x-4*ln(x^n)/d^5*e*ln(x)-1/3*n*(-6/d^5*e*ln(x)^2+12/d^ 
5*e*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))-4/d^4*e/(e*x+d)+13/d^5*e*ln(e*x+d 
)-1/2/d^3/(e*x+d)^2*e+3/d^4/x-13/d^5*e*ln(x)))-1/3*b^2*ln(x^n)^2/d^2/(e*x+ 
d)^3*e-3*b^2*e*n^2*ln(x)/d^5+3*b^2*e*n^2*ln(e*x+d)/d^5-8*b^2*e*n^2*poly...
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4} x^{2}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^2/x^2/(e*x+d)^4,x, algorithm="fricas")
 

Output:

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^4*x^6 + 4*d*e^3*x^ 
5 + 6*d^2*e^2*x^4 + 4*d^3*e*x^3 + d^4*x^2), x)
 

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{2} \left (d + e x\right )^{4}}\, dx \] Input:

integrate((a+b*ln(c*x**n))**2/x**2/(e*x+d)**4,x)
 

Output:

Integral((a + b*log(c*x**n))**2/(x**2*(d + e*x)**4), x)
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4} x^{2}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^2/x^2/(e*x+d)^4,x, algorithm="maxima")
 

Output:

-1/3*a^2*((12*e^3*x^3 + 30*d*e^2*x^2 + 22*d^2*e*x + 3*d^3)/(d^4*e^3*x^4 + 
3*d^5*e^2*x^3 + 3*d^6*e*x^2 + d^7*x) - 12*e*log(e*x + d)/d^5 + 12*e*log(x) 
/d^5) + integrate((b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + 2*(b^2*l 
og(c) + a*b)*log(x^n))/(e^4*x^6 + 4*d*e^3*x^5 + 6*d^2*e^2*x^4 + 4*d^3*e*x^ 
3 + d^4*x^2), x)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4} x^{2}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^2/x^2/(e*x+d)^4,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^2/((e*x + d)^4*x^2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2\,{\left (d+e\,x\right )}^4} \,d x \] Input:

int((a + b*log(c*x^n))^2/(x^2*(d + e*x)^4),x)
 

Output:

int((a + b*log(c*x^n))^2/(x^2*(d + e*x)^4), x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx=\frac {3 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e^{4} x^{6}+4 d \,e^{3} x^{5}+6 d^{2} e^{2} x^{4}+4 d^{3} e \,x^{3}+d^{4} x^{2}}d x \right ) b^{2} d^{8} x +9 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e^{4} x^{6}+4 d \,e^{3} x^{5}+6 d^{2} e^{2} x^{4}+4 d^{3} e \,x^{3}+d^{4} x^{2}}d x \right ) b^{2} d^{7} e \,x^{2}+9 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e^{4} x^{6}+4 d \,e^{3} x^{5}+6 d^{2} e^{2} x^{4}+4 d^{3} e \,x^{3}+d^{4} x^{2}}d x \right ) b^{2} d^{6} e^{2} x^{3}+3 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{e^{4} x^{6}+4 d \,e^{3} x^{5}+6 d^{2} e^{2} x^{4}+4 d^{3} e \,x^{3}+d^{4} x^{2}}d x \right ) b^{2} d^{5} e^{3} x^{4}+6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{6}+4 d \,e^{3} x^{5}+6 d^{2} e^{2} x^{4}+4 d^{3} e \,x^{3}+d^{4} x^{2}}d x \right ) a b \,d^{8} x +18 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{6}+4 d \,e^{3} x^{5}+6 d^{2} e^{2} x^{4}+4 d^{3} e \,x^{3}+d^{4} x^{2}}d x \right ) a b \,d^{7} e \,x^{2}+18 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{6}+4 d \,e^{3} x^{5}+6 d^{2} e^{2} x^{4}+4 d^{3} e \,x^{3}+d^{4} x^{2}}d x \right ) a b \,d^{6} e^{2} x^{3}+6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{e^{4} x^{6}+4 d \,e^{3} x^{5}+6 d^{2} e^{2} x^{4}+4 d^{3} e \,x^{3}+d^{4} x^{2}}d x \right ) a b \,d^{5} e^{3} x^{4}+12 \,\mathrm {log}\left (e x +d \right ) a^{2} d^{3} e x +36 \,\mathrm {log}\left (e x +d \right ) a^{2} d^{2} e^{2} x^{2}+36 \,\mathrm {log}\left (e x +d \right ) a^{2} d \,e^{3} x^{3}+12 \,\mathrm {log}\left (e x +d \right ) a^{2} e^{4} x^{4}-12 \,\mathrm {log}\left (x \right ) a^{2} d^{3} e x -36 \,\mathrm {log}\left (x \right ) a^{2} d^{2} e^{2} x^{2}-36 \,\mathrm {log}\left (x \right ) a^{2} d \,e^{3} x^{3}-12 \,\mathrm {log}\left (x \right ) a^{2} e^{4} x^{4}-3 a^{2} d^{4}-18 a^{2} d^{3} e x -18 a^{2} d^{2} e^{2} x^{2}+4 a^{2} e^{4} x^{4}}{3 d^{5} x \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:

int((a+b*log(c*x^n))^2/x^2/(e*x+d)^4,x)
 

Output:

(3*int(log(x**n*c)**2/(d**4*x**2 + 4*d**3*e*x**3 + 6*d**2*e**2*x**4 + 4*d* 
e**3*x**5 + e**4*x**6),x)*b**2*d**8*x + 9*int(log(x**n*c)**2/(d**4*x**2 + 
4*d**3*e*x**3 + 6*d**2*e**2*x**4 + 4*d*e**3*x**5 + e**4*x**6),x)*b**2*d**7 
*e*x**2 + 9*int(log(x**n*c)**2/(d**4*x**2 + 4*d**3*e*x**3 + 6*d**2*e**2*x* 
*4 + 4*d*e**3*x**5 + e**4*x**6),x)*b**2*d**6*e**2*x**3 + 3*int(log(x**n*c) 
**2/(d**4*x**2 + 4*d**3*e*x**3 + 6*d**2*e**2*x**4 + 4*d*e**3*x**5 + e**4*x 
**6),x)*b**2*d**5*e**3*x**4 + 6*int(log(x**n*c)/(d**4*x**2 + 4*d**3*e*x**3 
 + 6*d**2*e**2*x**4 + 4*d*e**3*x**5 + e**4*x**6),x)*a*b*d**8*x + 18*int(lo 
g(x**n*c)/(d**4*x**2 + 4*d**3*e*x**3 + 6*d**2*e**2*x**4 + 4*d*e**3*x**5 + 
e**4*x**6),x)*a*b*d**7*e*x**2 + 18*int(log(x**n*c)/(d**4*x**2 + 4*d**3*e*x 
**3 + 6*d**2*e**2*x**4 + 4*d*e**3*x**5 + e**4*x**6),x)*a*b*d**6*e**2*x**3 
+ 6*int(log(x**n*c)/(d**4*x**2 + 4*d**3*e*x**3 + 6*d**2*e**2*x**4 + 4*d*e* 
*3*x**5 + e**4*x**6),x)*a*b*d**5*e**3*x**4 + 12*log(d + e*x)*a**2*d**3*e*x 
 + 36*log(d + e*x)*a**2*d**2*e**2*x**2 + 36*log(d + e*x)*a**2*d*e**3*x**3 
+ 12*log(d + e*x)*a**2*e**4*x**4 - 12*log(x)*a**2*d**3*e*x - 36*log(x)*a** 
2*d**2*e**2*x**2 - 36*log(x)*a**2*d*e**3*x**3 - 12*log(x)*a**2*e**4*x**4 - 
 3*a**2*d**4 - 18*a**2*d**3*e*x - 18*a**2*d**2*e**2*x**2 + 4*a**2*e**4*x** 
4)/(3*d**5*x*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))